On certain extremal problems for polynomials
The authors use a result of the first author [J.Approximation Theory 26, 293-300 (1979; Zbl 0425.41010)] which states that for every \(\nu = (\nu_1, \dots, \nu_n) \in \mathbb{N}^n\) and \(h = (h_0, \dots, h_n) \in \mathbb{R}^{n + 1}\) with positive components there exists a unique polynomial \(\tau (\nu, h;x) : = c \prod^n_{j = 1} (x - x^*_j)^{\nu j} ( - 1 < x^*_1 < \cdots < x^*_n < 1)\) together with \(- 1 = t^*_0 < t^*_1 < \cdots < t^*_n = 1\) such that \(\tau (\nu, h, t^*_k) = ( - 1)^{\sigma_k} h_k\), \(0 \leq k \leq n\), \(\tau' (\nu, h;t^*_k) = 0\), \(0 < k < n\) hold, where \(\sigma_n : = 0\) and \(\sigma_k : = \nu_{k + 1} + \cdots + \nu_n\), \(0 \leq k < n\). In particular, there exist a unique monic polynomial \(\tau_* (\nu; x): = \prod^n_{j = 1} (x - x^*_j)^{\nu_j}\) and \(t^*_j\), \(0 \leq j \leq n\) (as above) with \(\tau_* (\nu; t^*_k) = ( - 1)^{\sigma_k} |\tau_* (\nu; \cdot) |_\infty\), \(0 \leq k \leq n\), where the usual \(L_p\)-norm on \([- 1,1]\), \(1 \leq p \leq \infty\), is used. The polynomial \(\tau_* (\nu; x)\) of degree \(N : = \nu_1 + \cdots + \nu_n\) generalizes the Chebyshev polynomial \(T_N (x)\) in the sense that \(|\tau_* (\nu; x) |_\infty = \inf_{- 1 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq 1} |\prod^n_{j = 1} (x - x_j)^{\nu_j} |_\infty\). The authors prove the following results. Theorem 1. For every \(\nu = (\nu_1, \dots, \nu_n) \in \mathbb{N}^n \) and \(h_1 = (h_{1,0}, \dots, h_{1,n})\), \(h_2 = (h_{2,0}, \dots, h_{2,n})\) with \(0 \leq h_{1,j} \leq h_{ 2j}\), \(0 \leq j \leq n\), \(|\tau^{(k)} (\nu, h_1; \cdot) |_p \leq |\tau^{(k)} (\nu, h_2; \cdot ) |_p\) holds for each \(1 \leq p \leq \infty\) and \(1 \leq k \leq N\). For an arbitrary polynomial (*) \(P (z) : = c \prod^n_{j = 1} (z - x_j)^{\nu_j}\), \((- 1 \leq x_1 \leq \cdots \leq x_n \leq 1)\) this yields \(|P^{(k)} |_p \leq {|\tau_*^{(k)} (\nu; \cdot) |_p \over |\tau_*(\nu; \cdot) |_\infty} |P |_\infty\) for \(1 \leq p \leq \infty\), \(1 \leq k \leq N\). Theorem 2. For any polynomial (*) and all \(z \in \mathbb{C}\), \(|z |\geq 1\), \(|P^{(k)} (z) |\leq {|\tau_*^{(k)} (\nu; z) |\over |\tau_*(\nu; \cdot) |_\infty} |P |_\infty\) holds for \(0 \leq k \leq N\). For \(|z |> 1\) this inequality is strict unless \(P\) is a constant multiple of \(\tau_* (\nu; \cdot)\).
scientific article; zbMATH DE number 765099